Differentiable weather models allow atmospheric initial conditions to be directly optimized to shadow a known trajectory, which we take from a reanalysis dataset. Here, we extend this optimization to include the forecast model, GraphCast, using low-rank adaptation (LoRA). We compare model optimization with initial-condition (IC) optimization, and then joint optimization, where both are tuned simultaneously. When compared against a control forecast initialized with reanalysis data, LoRA and IC optimization individually produce similar improvements. We find similar results when IC and LoRA are jointly optimized across a spectrum of solutions where a tuning coefficient smoothly shifts the weight of the joint optimization from all-LoRA to all-IC; forecast improvement is not unique to the relative contributions of model and IC corrections. To better discriminate the source of error, we introduce a second model, Pangu-Weather, as a supervisor that gates IC updates during joint optimization. This gated optimization is the same as before, except that IC updates at each epoch are accepted only when they improve the forecasts of both models; otherwise, only the LoRA update is accepted. Results show that this cross-model constraint substantially improves agreement between the two models in their response to the same optimized ICs, narrowing the gap in mean forecast error reduction from 64% to 10%. Moreover, when used together, the LoRA and IC corrections from the gated optimization yield GraphCast forecast improvements similar to those from the ungated optimization. These results imply that model error that is encoded in the ungated ICs is shifted to the LoRA corrections via gating, yielding IC corrections that may provide better analysis estimates.

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